Advanced Research on Second-Order Variational Analysis with New Applications to Optimization, Control, and Practical Modeling

该项目致力于开发各种应用中出现的大型系统的数学分析和优化的先进工具。这样的系统包括非标准的优化相关和平衡问题，其数据在通常意义上可能是不可微的。除了开发新的数学知识，首席研究员和他的七个博士生谁参加了该项目，（包括四名来自数学科学领域代表性不足的学生）特别关注社会经济学，交通平衡，行为科学和水资源应用中出现的各种模型。该项目研究二阶变分分析，优化，和系统控制，主要是由应用中出现的问题引起的。它有条件地分为五个相互关联的部分。第一部分是关于二阶广义微分理论及其在非线性分析的最优化相关领域中的应用的公开问题。特别注意的是建设性的计算主要的二阶广义导数显着类的扩展实值函数，发挥了至关重要的作用，在随后的部分项目。第二部分致力于各种稳定性概念的二阶刻画，包括非多面体锥规划中的倾斜和完全稳定性，以及椭圆变分不等式。该项目的第三部分涉及变分系统中的关键乘数的研究，这些乘数在很大程度上是原始对偶优化算法收敛缓慢的原因。在这里，PI和他的合作者开发了新的牛顿型方法来解决非光滑优化和相关问题。第四部分致力于控制理论的新发展，包括反馈控制和稳定的常微分方程和偏微分方程系统。考虑的另一个主要议题是最优控制的非凸版本的清扫过程中起着关键作用，在随后的应用。其中一些应用是该项目第五部分的重点，PI及其合作者对受控平面人群运动模型的优化问题进行了全面研究，该模型在社会经济学和交通平衡方面得到了广泛认可。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is devoted to developing advanced tools of mathematical analysis and optimization of large-scale systems arising in various applications. Such systems include nonstandard optimization-related and equilibrium problems with data that may not be differentiable in the usual sense. Besides developing new mathematical knowledge, the principal investigator and his seven PhD students who participate in the project (including four students from underrepresented groups in the mathematical sciences) pay particular attention to a variety of models arising in applications from socioeconomics, traffic equilibria, behavioral science, and water resources.The project studies new topics in second-order variational analysis, optimization, and systems control that are largely motivated by problems arising in applications. It is conditionally divided into five interrelated parts. Part I concerns open problems in second-order generalized differential theory and its applications to optimization-related areas of nonlinear analysis. Particular attention is paid to constructive computations of major second-order generalized derivatives for remarkable classes of extended-real-valued functions which play a crucial role in the subsequent parts of the project. Part II is devoted to second-order characterizations of various stability notions, including tilt and full stability in nonpolyhedral conic programming, and elliptic variational inequalities. Part III of the project deals with the study of critical multipliers in variational systems that are largely responsible for the slow convergence of primal-dual algorithms of optimization. Here, the PI and his collaborators develop new Newton-type methods to solve nonsmooth optimization and related problems. Part IV is devoted to novel developments in control theory, including feedback control and stabilization of ODE and PDE systems. Another major topic considered is optimal control of nonconvex versions of the sweeping process that plays a key role in subsequent applications. Some of these applications are the focus of Part V of the project, where the PI and his collaborators undertake a comprehensive study of the optimization problems for the controlled planar crowd motion model, which is well recognized in socioeconomics and traffic equilibria. Other types of applications considered stem from behavioral sciences and water resource models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.